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Abstract

We present a technique for large-scale optimization of optical microcavities based on the frequency-averaged local density of states (LDOS), which circumvents computational difficulties posed by previous eigenproblem-based formulations and allows us to perform full topology optimization of three-dimensional (3d) leaky cavity modes. We present theoretical results for both 2d and fully 3d computations in which every pixel of the design pattern is a degree of freedom (“topology optimization”), e.g. for lithographic patterning of dielectric slabs in 3d. More importantly, we argue that such optimization techniques can be applied to design cavities for which (unlike silicon-slab single-mode cavities) hand designs are difficult or unavailable, and in particular we design minimal-volume multi-mode cavities (e.g. for nonlinear frequency-conversion applications).

Figures (12)

Fig. 1 Starting with the naive objective of maximizing a microcavity’s Purcell factor Q/V, we perform a sequence of transformations of the problem in order to make it well posed and tractable. Here, we give a schematic diagram of each transformation, along with the corresponding section of the paper in which they are discussed.

Fig. 2 Contour integration path. The frequency-averaged LDOS is the path integral along arc A1 in the limit of an infinite-radius arc. By choosing the proper window/weight function W (ω) for optimizing LDOS in a desired bandwidth, the contribution along arcs A2 and A3 can be made negligible compared to A1. Therefore, the residues at poles ω̄k enclosed by this contour can be used to approximate the averaged LDOS.

Fig. 3 For 2d cavity optimization, we start in Secs. 8.1–8.3 by optimizing over every pixel in the interior of the computational domain as indicated in (a). This leads to cavities that utilize bandgap structures to confine light with arbitrary Q, regardless of V, limited only by the size of the domain. In order to investigate Q vs. V tradeoffs analogous to those in 3d slabs, in Sec. 8.4 we limit the degrees of freedom to a thin strip (b), which imposes intrinsic radiation losses (perpendicular to the strip) and forces the optimization to sacrifice V in order to increase Q. A full 3d optimization is considered in Sec. 8.5.

Fig. 9 2D TE optimization for thin strips with fixed width (geometry sketched in Fig. 3b). Fig. (a): Qrad vs. Q̃ for 2d thin stirps with same width (λ) but different length (d = λ, 2λ, 3λ, 5λ). As Q̃ is increased in the optimization, higher Qrad are obtained until Qrad is limited by the degrees of freedom. As the degrees of freedom increase, Qrad first gets bigger, but becomes saturated at some level around 107 due to numerical precision. Fig. (b): An optimized 2d thin-strip structure with width λ and length d = 5λ.

Fig. 10 We wish to optimize a microcavity in an air-membrane Si slab in Sec. 8.5, with the effective computational domain depicted in (a), where the degrees of freedom are every pixel in the 2d pattern of the slab cross-section for a fixed thickness. Since all 2d single-polarization optimizations found structures with two mirror symmetry planes, we can reduce the computational domain to 1/8 the volume (b) by imposing these mirror symmetries along with the vertical mirror symmetry.